2019 AMC 10A Problems/Problem 15

The following problem is from both the 2019 AMC 10A #15 and 2019 AMC 12A #9, so both problems redirect to this page.

Problem

A sequence of numbers is defined recursively by $a_1 = 1$ , $a_2 = \frac{3}{7}$ , and $a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$ for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$

$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

Solution 1

Using the recursive formula, we find $a_3=\frac{3}{11}$ , $a_4=\frac{3}{15}$ , and so on. It appears that $a_n=\frac{3}{4n-1}$ , for all $n$ . Setting $n=2019$ , we find $a_{2019}=\frac{3}{8075}$ , so the answer is $\boxed{\textbf{(E) }8078}$ .

To prove this formula, we use induction. We are given that $a_1=1$ and $a_2=\frac{3}{7}$ , which satisfy our formula. Now assume the formula holds true for all $n\le m$ for some positive integer $m$ . By our assumption, $a_{m-1}=\frac{3}{4m-5}$ and $a_m=\frac{3}{4m-1}$ . Using the recursive formula, $a_{m+1}=\frac{a_{m-1}\cdot a_m}{2a_{m-1}-a_m}=\frac{\frac{3}{4m-5}\cdot\frac{3}{4m-1}}{2\cdot\frac{3}{4m-5}-\frac{3}{4m-1}}=\frac{(\frac{3}{4m-5}\cdot\frac{3}{4m-1})(4m-5)(4m-1)}{(2\cdot\frac{3}{4m-5}-\frac{3}{4m-1})(4m-5)(4m-1)}=\frac{9}{6(4m-1)-3(4m-5)}=\frac{3}{4(m+1)-1},$ so our induction is complete.

Solution 2

Since we are finding the sum of the numerator and the denominator, consider the function $b_n = \frac{1}{a_n}$ .

$\frac{1}{a_n} = \frac{2a_{n-2}-a_{n-1}}{a_{n-2} \cdot a_{n-1}}=\frac{2}{a_{n-1}}-\frac{1}{a_{n-2}}$

$b_n = 2b_{n-1}-b_{n-2}=3b_{n-2}-2b_{n-3}=4b_{n-3}-3b_{n-4}=\dots$

By recursively following this pattern, we can see that $b_n=(n-1) \cdot b_2 - (n-1) \cdot b_1$ .

By plugging in 2019, we get: $b_{2019} = 2018 \cdot \frac{7}{3}-2017 = \frac{8705}{3}$ . Since the numerator and the denominator are relatively prime, the answer is $8708$ which implies $\rightarrow \boxed{E}$

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 10 Problems and Solutions

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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2019 AMC 10A Problems/Problem 15

Contents

Problem

Solution 1

Solution 2

See Also